Frequentist Statistics
Bayesian
nls functionnls the goal was just to find the “BEST” set of parametersRhat compares variance between and within chains. Should be < 1.05ESS (Effective sampling size) should be > 100 per chainPlease create two objects
datsmall that only retains observations from “2023-01-27”datbcd that only retains BH, CC, and DPR observations from “2023-01-27” prior class coef group resp dpar nlpar lb ub
(flat) b
(flat) b popCC
(flat) b popDPR
student_t(3, 3.5, 2.5) Intercept
student_t(3, 0, 2.5) sigma 0
source
default
(vectorized)
(vectorized)
default
default
threads = 4 means that we want each chain to run in parallel in a different CPU in our computer.m1 <- brm(height_cm ~ pop,
data = datbcd,
prior = set_prior("normal(0,10)", class = "b"),
sample_prior = TRUE,
threads = 4)Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 13.1.6 (clang-1316.0.21.2.5)’
using SDK: ‘MacOSX12.3.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 "-I/opt/R/arm64/include -I/opt/homebrew/include" -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:88:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
namespace Eigen {
^
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
namespace Eigen {
^
;
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
#include <complex>
^~~~~~~~~
3 errors generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000171 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.71 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
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Chain 1:
Chain 1: Elapsed Time: 0.06 seconds (Warm-up)
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Chain 1: 0.111 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 6e-06 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.06 seconds.
Chain 2: Adjust your expectations accordingly!
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Chain 2:
Chain 2: Elapsed Time: 0.055 seconds (Warm-up)
Chain 2: 0.053 seconds (Sampling)
Chain 2: 0.108 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 9e-06 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.09 seconds.
Chain 3: Adjust your expectations accordingly!
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Chain 3:
Chain 3: Elapsed Time: 0.071 seconds (Warm-up)
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Chain 3: 0.131 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 1.3e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
Chain 4: Adjust your expectations accordingly!
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Chain 4:
Chain 4: Elapsed Time: 0.091 seconds (Warm-up)
Chain 4: 0.096 seconds (Sampling)
Chain 4: 0.187 seconds (Total)
Chain 4:
The output shows us the progress of each chain
Family: gaussian
Links: mu = identity; sigma = identity
Formula: height_cm ~ pop
Data: datbcd (Number of observations: 155)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 3.22 0.12 2.98 3.47 1.00 3679 2887
popCC 1.45 0.25 0.96 1.93 1.00 4365 3309
popDPR 1.89 0.31 1.29 2.50 1.00 4094 3185
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.25 0.07 1.12 1.41 1.00 4271 2666
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
Estimate is the estimated height
Intercept is for the reference population (BH in this case)popCC is the estimated difference between CC and BHRhat and the two ESS stats. What did we want these to be?Our posterior plots had two coefficients that we did not specify priors for. What was used?
prior class coef group resp dpar nlpar lb ub
normal(0,10) b
normal(0,10) b popCC
normal(0,10) b popDPR
student_t(3, 3.5, 2.5) Intercept
student_t(3, 0, 2.5) sigma 0
source
user
(vectorized)
(vectorized)
default
default
Post_prob to be near 1 (probability that hypothesis is true)Evid.Ratio to > 3 (>3 = “moderate evidence”, > 10 = “strong evidence”)Hypothesis Tests for class b:
Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
1 (popCC) > 0 1.45 0.25 1.04 1.86 Inf 1 *
---
'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
'*': For one-sided hypotheses, the posterior probability exceeds 95%;
for two-sided hypotheses, the value tested against lies outside the 95%-CI.
Posterior probabilities of point hypotheses assume equal prior probabilities.
Sometime I prefer to do this by “hand”
[1] 4000 12
# A draws_df: 6 iterations, 1 chains, and 9 variables
b_Intercept b_popCC b_popDPR sigma prior_Intercept prior_b prior_sigma lprior
1 3.2 0.74 1.9 1.3 3.42 16.5 3.920 -9.8
2 3.4 1.34 1.4 1.2 -0.48 -10.0 4.444 -9.8
3 3.1 1.79 1.8 1.3 3.09 16.4 3.284 -9.8
4 3.3 1.10 1.9 1.2 2.80 -4.8 1.350 -9.8
5 3.2 1.50 2.1 1.3 -1.50 -9.1 3.076 -9.8
6 3.2 1.44 2.0 1.4 3.85 2.8 0.077 -9.8
# ... with 1 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}
# A tibble: 1 × 1
CC_greater_than_zero
<dbl>
1 1
Post_prob to be near 1 (probability that hypothesis is true)Evid.Ratio to > 3 (>3 = “moderate evidence”, > 10 = “strong evidence”)Hypothesis Tests for class b:
Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
1 (popCC)-(popDPR) = 0 -0.44 0.36 -1.15 0.28 19.59
Post.Prob Star
1 0.95
---
'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
'*': For one-sided hypotheses, the posterior probability exceeds 95%;
for two-sided hypotheses, the value tested against lies outside the 95%-CI.
Posterior probabilities of point hypotheses assume equal prior probabilities.
m2 <- brm(height_cm ~ pop + (1|block),
data = datbcd,
prior = set_prior("normal(0,10)", class = "b"),
sample_prior = TRUE,
threads = 4)Running /Library/Frameworks/R.framework/Resources/bin/R CMD SHLIB foo.c
using C compiler: ‘Apple clang version 13.1.6 (clang-1316.0.21.2.5)’
using SDK: ‘MacOSX12.3.sdk’
clang -arch arm64 -I"/Library/Frameworks/R.framework/Resources/include" -DNDEBUG -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/Rcpp/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/unsupported" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/BH/include" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/src/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppParallel/include/" -I"/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 "-I/opt/R/arm64/include -I/opt/homebrew/include" -fPIC -falign-functions=64 -Wall -g -O2 -c foo.c -o foo.o
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:88:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:1: error: unknown type name 'namespace'
namespace Eigen {
^
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:628:16: error: expected ';' after top level declarator
namespace Eigen {
^
;
In file included from <built-in>:1:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22:
In file included from /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Dense:1:
/Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/library/RcppEigen/include/Eigen/Core:96:10: fatal error: 'complex' file not found
#include <complex>
^~~~~~~~~
3 errors generated.
make: *** [foo.o] Error 1
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 0.000185 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.85 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
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Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 1:
Chain 1: Elapsed Time: 0.34 seconds (Warm-up)
Chain 1: 0.352 seconds (Sampling)
Chain 1: 0.692 seconds (Total)
Chain 1:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 6e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.6 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
Chain 2:
Chain 2: Iteration: 1 / 2000 [ 0%] (Warmup)
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Chain 2:
Chain 2: Elapsed Time: 0.43 seconds (Warm-up)
Chain 2: 0.364 seconds (Sampling)
Chain 2: 0.794 seconds (Total)
Chain 2:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 2.7e-05 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.27 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
Chain 3:
Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup)
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Chain 3: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 3:
Chain 3: Elapsed Time: 0.537 seconds (Warm-up)
Chain 3: 0.291 seconds (Sampling)
Chain 3: 0.828 seconds (Total)
Chain 3:
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 2.5e-05 seconds
Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.25 seconds.
Chain 4: Adjust your expectations accordingly!
Chain 4:
Chain 4:
Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
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Chain 4: Iteration: 2000 / 2000 [100%] (Sampling)
Chain 4:
Chain 4: Elapsed Time: 0.482 seconds (Warm-up)
Chain 4: 0.39 seconds (Sampling)
Chain 4: 0.872 seconds (Total)
Chain 4:
prior class coef group resp dpar nlpar lb ub
normal(0,10) b
normal(0,10) b popCC
normal(0,10) b popDPR
student_t(3, 3.5, 2.5) Intercept
student_t(3, 0, 2.5) sd 0
student_t(3, 0, 2.5) sd block 0
student_t(3, 0, 2.5) sd Intercept block 0
student_t(3, 0, 2.5) sigma 0
source
user
(vectorized)
(vectorized)
default
default
(vectorized)
(vectorized)
default
Family: gaussian
Links: mu = identity; sigma = identity
Formula: height_cm ~ pop + (1 | block)
Data: datbcd (Number of observations: 155)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Group-Level Effects:
~block (Number of levels: 10)
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept) 0.50 0.20 0.19 0.98 1.00 1212 1306
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept 3.23 0.21 2.83 3.64 1.00 1567 1714
popCC 1.41 0.24 0.95 1.90 1.00 4199 2709
popDPR 1.85 0.28 1.28 2.41 1.00 4213 2675
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma 1.18 0.07 1.05 1.33 1.00 3875 2936
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
elpd_diff se_diff
m2 0.0 0.0
m1 -5.4 3.3
elpd_diff is how much worse a model is relative to the preferred modelse_diff to favor the more complex model